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Statistics Ch. 7, 8, 9, 10, 11
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Terms in this set (179)
If the sampling distribution of x is normal, we can transform x into the standard normal random variable as
z = xu/o/SqRtn
Match the term on the left with its formula on the right:
The standard deviation of the sample mean=o/SqRtn
The expected value of the sample mean=u
The variance of the sample mean=o^2/n
A control chart is a plot of calculated statistics of a production process over _____.
time
A sample of n observations that have the same probability of being selected from the population as any other sample of n observations is called a(n) _____.
simple random sample
Match the definitions with the terms:
Parameter = constant
Statistic = random variable
Estimate = a particular value of an estimator
If p is the value that a normal random variable assumes, then we can transform it into its standard normal value as

z = pp/SqRtp(1p)/n
When a firm applies statistical techniques to develop and maintain its ability to produce highquality goods and services, it is implementing statistical _____.
quality control
Bias can occur in sampling. Bias refers to
the tendency of a sample statistic to systematically over or underestimate a population parameter.
The central limit theorem states that, for any distribution, as n gets larger, the sampling distribution of the sample mean becomes
closer to normal distribution.
The general rule for using the finite correction factor is that the sample constitutes at least _____ of the population.
5%
In a statistical problem, a population consists of
all items of interest
Which of the following is an example of a sample statistic?
_
x
_
The variance of x, which is equal to o^2 / n, is
smaller than the variance of the individual observation o^2.
What is a primary requirement for a "good" sample?
It is representative of the population we are trying to describe.
_
For an x chart, control limits are calculated as
u +or 3xo/SqRtn
The probability distribution of the sample mean is commonly referred to as the _____.
_
sampling distribution of X.
_
The expected value of p is the
proportion of successes in the population.
The branch of statistics that uses sample statistics to estimate a population parameter or test a hypothesis about such a parameter is BEST referred to as _____.
inferential statistics
Selection bias occurs when
there is a systematic exclusion of certain groups from consideration for the sample.
_
The control limits for a p chart are defined as
p +or SqRtp(1p)/n.
When a sample statistic is used to make inferences about a population parameter, it is referred to as a/an
estimator.
Random samples of size 100 are taken from a population whose population proportion is 0.40. The expected value of the sample proportion is
0.40
Assignable variation is caused by
specific events that can usually be identified and eliminated.
The sample size required to approximate the normal distribution depends on
how much the population varies from normality.
When the finite population correction factor is applied to the sample proportion, the resulting standard deviation for the sample mean is equal to
SqRtp(1p)/n x SqRt(Nn)/N1).
In general, the two approaches that are used for statistical quality control are
acceptance sampling and the detection approach.
A population has a mean of 100 and a standard deviation of 12. A random sample of 36 is selected. The standard
_
deviation of x is equal to
2 _
The standard deviation of x is equal to o/SqRtn =12/SqRt36 = 2
A sporting goods manufacturer wants to ensure that the balls it produces have a diameter of 30 inches. The standard deviation is 0.5 inches. The manufacturer samples 25 balls every 5 hours and calculates the mean diameter. The lower control limit is _____ inches.
29.7
103(0.5/SqRt25)
True or false: Chance variation is not usually under the control of an individual worker or machine.
True
_
The standard deviation of p equals.
SqRtp(1p)/n
A population has a mean of 100 and a standard deviation of 10. A random sample of 25 is selected. The expected value
_
of x is equal to
100 _
The expected value of x is equal to u=100
Stratified sampling is preferred to cluster sampling when the objective is
to increase precision
A manufacturing production process is in control if the sample means are
randomly spread out between the control limits.
When the finite population correction factor is applied to the sample mean, the resulting standard deviation for the sample mean is equal to
(o/SqRtn) SqRt(Nn)/(N1).
Random samples of size 400 are taken from a population whose population proportion is 0.25. The expected value of the sample proportion is
0.25
A population has a mean of 50 and a standard deviation of 10. A random sample of 144 is selected. The expected value
_
of x is equal to
_
The expected value of x is equal to u=50
***
***
When the confidence level increases from 95% to 99%, the confidence interval for the population mean _____.
widens
A confidence interval narrows if the following is accomplished:
the chosen confidence level decreases.
the sample size increases
True or false: A 95% confidence interval for u implies that if numerous samples are taken from a population, 95% of the intervals will contain u.
True
Assume the sample proportion is equal to 0.70 in a sample size of 100. In addition, z0.05=1.645 and t0.05,99=1.660. A 90% confidence interval for the population proportion is
0.70+or1.645 SqRt0.70(10.70)/100
The most practical way to reduce the margin of error is to _____.
increase the sample size
True of false: A confidence interval provides a range of values that should contain a population parameter with a certain level of confidence.
True
True of false: For a given sample size n and a population standard deviation o, the lower the confidence level 100(1a)%, the narrower the confidence interval.
True
Requiring less confidence results in a narrower interval.
The parameter p represents the
population proportion
When the population standard deviation is unknown, the standard error for the sample mean is calculated as
s/SqRtn.
In order to construct a confidence interval for u, the sampling distribution of the estimator x* must follow or approximately follow a(n) _____ distribution.
normal
A 95% confidence interval for the population mean is calculated as (40, 80). The margin of error for this interval is _____.
20
First find the point estimate (40+80)/2=60. Then the margin of error can be found as 8060=20
The sampling distribution of estimator x* follows a normal distribution when the underlying population normally distributed and/or when the sample size is large enough. As a ruleofthumb, we use the following:
n>or=30
When constructing a confidence interval for the population mean when the population standard deviation is unknown, the degrees of freedom for the t distribution are defined as
n1
A 95% confidence interval for the population mean is constructed as 6+or2. What is the probability of error, α?
0.05
A random sample of 60 observations results in 42 successes. What is the point estimate of the population proportion of successes?
0.7
Which of the following is the correct formula for the margin of error in the interval estimation of p?
Zα/2SqRtp
(1p
*)/n
A sample of 25 is drawn from normal population with a population standard deviation of 100. A sample mean of 35 is calculated. For a 95% confidence interval z0.025=1.96. A 95% confidence interval for the population mean is equal to
35+or1.96x100/SqRt25
All of the following are components of the formula for selectin n to estimate p EXCEPT:
o*
Which of the following is a descriptive measure for a qualitative variable?
Proportion
What is the confidence level if α=0.10
90%
100(10.10)
For a desired margin of error E, the minimum sample size required to estimate a 100(1α)% confidence interval for the population mean is
n=(zα/2o*/E)^2
Whenever we construct a confidence interval for the population mean, the margin of error includes the standard error of x* and the
desired level of confidence
What is the value of zα/2 for a 99% confidence interval for the population mean?
2.576
z0.01/2= z0.005
If repeated samples of size n are taken from a normal population with an unknown variance, then the statistic _____ follows the t distribution with n1 degrees of freedom.
T=x*u/s/SqRtn
Suppose you are constructing a confidence interval for the population mean. For a given confidence level and standard deviation, the width of the interval is wider for a
smaller sample size
The allowed probability that an interval estimate of a population mean will not contain u is referred to as the
level of significance.
Suppose you are constructing a confidence interval for the population mean. For a given sample size and population standard deviation, how will the width of the interval change as the confidence level increases?
It gets larger
Suppose you are constructing a confidence interval for the population mean. For a given confidence level and sample size, the width of the interval is wider for a
larger standard deviation
AAA batteries are advertised to have a life of about 9 hours of use. With a certain level of confidence, it is advertised that the life is between 810 hours. If 9 hours is the point estimate, then the margin of error is
1 hour
Each t distribution is identified by its
degrees of freedom
What is the value of zα/2 for a 90% confidence interval for the population mean?
1.645
When comparing two confidence intervals, the one that has the smaller margin of error has a
more precise estimate of the parameter.
When the sample size is sufficiently large, we can approximate the sampling distribution of the sample proportion using the
normal distribution
The confidence coefficient equals
1α
The standard error of the sample mean is NOT affected by the
confidence level.
The confidence level affects the margin of error, but not the standard error.
The probability of error α for a 90% confidence interval is _____ and the probability of error α for a 99% confidence interval is _____.
0.10, 0.01
Precision in interval estimates is implied by a(n) _____ margin of error.
low
***
***
The two competing hypotheses used in hypothesis testing are called the _____ hypothesis and the _____ hypothesis.
null, alternative
In inferential statistics, we use _____ information to make inferences about an unknown _____ parameter.
sample, population
Put the following steps in the pvalue approach to hypothesis testing in the correct order.
1. Specify the null and alternative hypotheses.
2. Specify the significance level.
3.Calculate the value of the test statistic and its pvalue.
4. State the conclusion and interpret results.
The two equivalent methods to solve a hypothesis test are the
pvalue approach
critical value approach
For a hypothesis test concerning the population proportion p, the value of the test statistic is calculated as
z=p*p0/SqRtp0(1p0)/n
For an alternative hypothesis of HA:u>u0, we might possibly reject the null hypothesis if
the sample means is greater than u0
Specify the competing hypotheses that would be used in order to determine whether the population mean differs from 15.
H0: u=15 versus HA: u does not = 15
If a 95% confidence interval for the mean value of a store's customer accounts is computed as $850 + or  70, then the null hypothesis of a twotailed hypothesis test would be rejected if the value of u0 is less than$ _____ or greater than $ _____.
$780, $920
Suppose the competing hypotheses for a test are H0: u=40 versus HA: u does not = 40. The value of the test statistic is t29 = 2.22 with a corresponding 2tailed pvalue of 0.0344, then at the 10% level of significance the correct conclusion is:
Reject H0 and conclude that the population mean appears to differ from 40.
When testing u, the pvalue is the probability of obtaining a sample mean at least as large or at least as small as the one derived from a given sample, assuming the _____ hypothesis is true.
null
Suppose you are performing a hypothesis test on u and the value of o is known. At the 10% significance level, the critical value(s) for a lefttailed test is (are):
z0.10
A twotailed test of the population mean is conducted at α = 0.10. The calculated test statistic is z=1.55 and P(Z> or = 1.55)=0.0606. The null hypothesis should _____.
not be rejected since the pvalue = 0.1212>0.10.
Suppose the competing hypotheses for a test are H0: u=10 versus HA: u does not =10. If the value of the test statistic is 1.87 and the critical values at the 5% level of significance are z0.025=1.96 and z0.025=1.96, then the correct conclusion is:
Do not reject H0 and conclude that the population mean does not appear to differ from 10 at the 5% significance level.
In hypothesis testing, two incorrect decisions are possible:
Not rejecting the null hypothesis when it is false.
Rejecting the null hypothesis when it is true.
Suppose the competing hypotheses for a test are H0:p< or = 0.75 versus HA:p>0.75. If the value of the test statistic is z=2.78, with a corresponding pvalue=0.0270, then at a significance level of 0.05 the correct conclusion is:
Reject H0 and conclude that the population proportion appears to be greater than 0.75.
Since the pvalue = 0.0270 < 0.05 = α, we reject the null hypothesis and conclude that the population proportion is greater than 0.75.
We do not reject the null hypothesis when the pvalue is
> or = α
An auditor for a small company suspects that the mean customer account balances have fallen below $550 per month, the average amount for all customer accounts over the past 5 years. She takes a random sample of 40 accounts and computes the sample mean as $543. State the hypotheses for testing the auditor's claim.
H0:u> or = 550 and HA:u<550
For a hypothesis test on u when the value of o is unknown, the value of the test statistic is calculated as _____, provided that we sample from a normal population.
tdf= x*u/s/SqRtn
For a hypothesis test of u when o is known, the value of the test statistic is calculated as
z=x*u0/o/SqRtn
The following image of a two tailed test illustrates that if z>0, then the pvalue is equal to
2xP(X> or = z).
Suppose the competing hypotheses for a test are H0:u=100 versus HA:u does not = 100. If the pvalue for the hypothesis test is 0.07 and the chosen level of significance is 0.05, then the correct conclusion is:
Do not reject H0 and conclude that the population mean does not differ from 100 at the 5% significance level.
Suppose the competing hypotheses for a test are H0:p> or = 0.45 versus HA:p<0.45. If the value of the test statistic is z=1.35, with a corresponding pvalue = 0.0885, then at the 1% level of significance the correct conclusion is:
Do not reject H0 and conclude that the population proportion does not appear to be less than 0.45.
True of false: The alternative hypothesis typically states the opposite of the null hypothesis.
True
A Type I error occurs when we
Reject the null hypothesis when it is actually true.
Unlike the mean and standard deviation, the population proportion p is a descriptive summary measure that can be used for data that is
qualitative
The normal distribution approximation for a binomial distribution is valid when
np> or =5 and n(1p)> or =5
True of false: We choose a value for a before conducting a hypothesis test.
True
The expected value of the sampling distribution of P* is the _____.
Population proportion
Which of the following statements is NOT correct concerning the pvalue and critical value approaches to hypothesis testing?
Both approaches use the same decision rule concerning when to reject H0.
In hypothesis testing, if the sample data provides significant evidence that the null hypothesis is incorrect, then we
reject the null hypothesis
Which of the following is true?
a = the probability of committing a Type I error; B = the probability of committing a Type II error.
True or false: In the critical value approach, if the value of the test statistic does not fall within the rejection region, then we reject the null hypothesis.
False
Which of the following is NOT a step we use when formulating the null and alternative hypotheses?
Calculate the value of the sample statistic.
We do not need this information when formulating the null and alternative hypotheses.
Hypothesis testing enables us to determine if the collected _____ data is inconsistent with what is stated in the null hypothesis.
sample
A quality control officer believes that the average time of use for AAA batteries differs from the claimed 8.5 hours. The QC officer takes a random sample of 30 AAA batteries and finds that the sample mean is 8.7 hours. State the null and alternative hypothesis for testing the officer's claim.
H0:u=8.5 and HA:u does not =8.5
The pvalue is calculated assuming the
null hypothesis is true
If the population standard deviation is unknown, it can be estimated by using
s.
In a hypothesis test, u0 and p0 are hypothesized values of the _____ mean and the _____ proportion, respectively.
population, population
True or false: For a given sample size n, a Type I error can only be reduced at the expense of a higher Type II error.
True
If the value of the test statistic falls in the rejection region, the pvalue must be
less than a
When testing u and o is known, H0 can never be rejected if z< or = 0 for a
righttailed test
A Type II error occurs when we
Do not reject the null hypothesis when it is actually false.
True or false: The optimal values of Type I and Type II errors require a compromise in balancing the costs of each type of error.
True
In general, the null and alternative hypotheses are
mutually exclusive
In hypothesis testing, the standard error of the sample proportion p* is computed as
SqRt p0(1p0)/n
The null hypothesis is specified by using one of the following signs:
=, <or=, >or=
***
***
The hypotheses to determine whether the average AAA battery life for Brand 1 differs from Brand 2 are
H0:u1u2=0 versus HA:u1u2 does not =0
Suppose the competing hypotheses for a test are H0: UD< or =0 versus HA: UD> or =0. If the value of the test statistic is t24=2.06 and the critical value at the 5% significance level is t0.05,24=1.711, then the correct conclusion is:
Reject H0; the mean difference is greater than zero.
Statistical inference concerning the mean difference based on matchedpairs sampling requires one of two conditions. Select the two conditions.
the sample size n>or=30
The paired difference D=X1X2 is normally distributed.
Performing a oneway ANOVA test, instead of performing a series of twosample t tests, _____ the risk of incorrectly rejecting the null hypothesis.
reduces
A confidence interval for the mean difference UD follows the general format of a point estimate +or
margin of error
The test statistic for a oneway ANOVA test is equal to
MSTR/MSE.
A particular personal trainer works primarily with track and field athletes. She believes that, on average, her clients run faster after going through her program for 6 weeks. How might she test that claim?
A matched pairs hypothesis test for UD.
True or false: The oneway ANOVA test is based on the Fdf1,df2 distribution.
TRUE
For matchedpairs sampling, the parameter of interest is referred to as the
mean difference
When examining the difference between two population means, if the populations cannot be assumed normal, then X
1X
2 is approximately normal only if
n1>or=30 and n2>or=30.
In most applications, the hypothesized difference between two population means is _____.
zero
In oneway ANOVA, one of the independent estimates of the common population variance is NOT based on
the population median
We calculate the _____ as a point estimate of an unknown population variance.
sample variance
Which of the following is NOT an assumption for performing a oneway ANOVA?
The population correlation coefficients indicate a strong linear relationship.
The oneway ANOVA test is always a
righttailed test
A pooled estimate of the common variance between two populations is used when
both populations are assumed to have the same population variance.
Since ANOVA techniques were originally developed in connection with agricultural experiments, the term _____ is often used to identify the populations being examined for an ANOVA analysis.
treatment
We calculate a _____ estimate of the common variance, denoted sp^2, when two populations are assumed to have the same variance.
pooled
A 95% confidence interval for the mean difference uD is 1.0 +or3.5, indicating that at the 5% significance level, the mean difference is
not significant and the mean difference does not differ from zero.
When constructing a confidence interval for the difference between two population means, the margin of error equals
the standard error multiplied by za/2 or ta/2,df*
The competing hypotheses for a oneway ANOVA test that compares the means of three populations are defined as
H0: u1=u2=u3
HA: Not all populations means are equal.
The confidence interval for the difference between two population means when the population standard deviations are unknown and assumed equal is
(x
1x
2)+orta/2,df SqRt sp^2(1/n^1 + 1/n^2)
Suppose the competing hypotheses for a test are H0:uD<or=0 versus HA:uD>0. If the value of the test statistic is t24=2.06 and the critical value at the 5% significance level is t0.05,24=1.711, then the correct conclusion is:
Reject H0; the mean difference is greater than zero.
Since t24=2.06>1.711=t, we reject the null hypothesis and conclude that the mean difference is greater than zero
Two or more samples are _____ if the process that generated one sample is completely separate from the process that generated the other sample.
independent
A matchedpairs sample looks to find
a natural pairing between one observation in the first sample and one observation in the second sample.
Suppose the competing hypotheses for a test are H0:u1=u2=u3 versus HA: Not all population means are equal. If the value of the test statistic is F2,14=5.75 and the critical value at the 5% level of significance is F0.05,2,14=3.74, then the correct conclusion is:
Reject H0; we conclude that some means differ.
True or false: ANOVA is a statistical technique used to determine if there is a difference in three or more population standard deviations.
False
An ANOVA test determines if differences exist between the mean of three or more populations.
In ANOVA testing, if the ratio of the betweentreatment variability to withintreatment variability is significantly greater than one, then we
reject the null hypothesis and conclude that not all population means are equal.
If the selection of one sample does not affect the selection of another sample, then the samples are considered _____.
independent
In order to determine if there is a difference between the means of 3 or more populations, we use
analysis of variance.
A specific type of dependent sampling when the samples are paired in some way is called
matchedpairs sampling
If we want to determine whether two population proportions differ by 25%, then the competing hypotheses would be
H0: p1p2=0.25 versus HA: 1p2 does not =.25
Consider a multinomial experiment with a null hypothesis of H0: p1=p2=p3=p4=0.25. The value of the test statistic is X^2 3 = 10.25. At the 5% significance level, the critical value is X^2 0.05,3=7.815. The correct decision is to:
reject H0: conclude that not all of the population proportions are equal to 0.25.
Since X^2 3 = 10.25 > 7.815 = X^2 0.05,3 we reject H0 and conclude that not all of the population proportions are equal to 0.25.
A researcher finds that 15 out of 45 customers in store 1 like type A cereal and 20 out of 40 customers in store 2 like type A cereal. The sample proportions are
0.33 for store 1 and 0.50 for store 2
Suppose the competing hypotheses for a test are H0: p1p2=0 versus HA: p1p2 does not =0. A 95% confidence interval for the difference between the population proportions is calculated as [0.25, 0.40]. At the 5% significance level, the correct conclusion to the hypothesis test is:
Reject H0; the population proportions are significantly different.
Since the confidence interval does not include d0 = 0, we reject the null hypothesis and conclude that the population proportions are significantly different.
Consider a test of independence with a null hypothesis of H0: Ethnicity and Arrest are independent. The value of the test statistic is X^2 3=8.25 and its pvalue is 0.0411. At the 5% significance level, the correct decision is to:
Reject H0; conclude that Ethnicity and Arrest are dependent.
Since the pvalue = 0.0411 < 0.05 = a, we reject H0 and conclude that Ethnicity and Arrest are dependent.
The test statistic for a test of independence compares each cell's _____.
observed frequency with expected frequency
A test of independence is also called a chisquare test of a
contingency table
True or false: A multinomial experiment can set each population proportion equal to a different predetermined hypothesized value so long as the population proportions sum to one.
True
Suppose a sociologist wants to determine whether the variables 'Race' and 'Arrest' are independent. The competing hypotheses take the following form:
H0: Race and Arrest are not dependent.
HA: Race and Arrest are dependent.
A multinomial experiment is a generalization of _____ experiment.
a binomial
A lawyer wants to show that gender and ability to be promoted are not related. She can do this by conducting a
test of independence
For the goodnessoffit test, the chisquare test statistic will always be
at least zero
For a multinomial experiment, which of the following is NOT accurate?
There are a series of n dependent trials.
Which of the following null hypotheses is used to test if five population proportions are the same?
H0: p1p2=p3=p4=p5=0.20
A binomial experiment is a series of n identical trials of a random experiment where each trial has two possible outcomes, labeled _____.
success and failure
Consider a test of independence with a null hypothesis of H0: Gender and Promotion are independent. The value of the test statistic is X^2 1=5.75. At the 5% significance level, the critical value is X^2 0.05,1=3.841. The correct decision is to:
reject H0: conclude that Gender and Promotion are dependent.
The confidence interval for difference between two population proportions is
(P
1P
2) +or za/2 SqRt P
1(1P
1)/n1 + P
2(1P
2)/n2.
For the chisquare test of a contingency table, the expected cell frequencies are computed as
the row total multiplied by the column total divided by the sample size.
Consider a multinomial experiment with a null hypothesis of H0: p1=0.40, p2=0.10, p3=0.35, p4=0.15. The value of the test statistic is X^2 3=9.65 and its pvalue is 0.0218. At the 5% significance level, the correct decision is to:
reject H0; conclude that not all population proportions are equal to their hypothesized values.
In a customer service survey, the service is rated as excellent, good, fair, or poor. This survey is an example of _____ experiment.
a multinomial
Which of the following null hypotheses is used to perform a multinomial test on four categories to determine whether four population proportions are all the same?
H0: p1=p2=p3=p4=0.25
The goodnessoffit test statistic is valid as long as the expected frequencies in each category are _____ or more.
5
A pooled sample proportion can be computed when testing to see if two population proportions are equal. The pooled value represents an estimate of the unknown _____.
population proportion
The degrees of freedom for a contingency table with four rows and three columns is
6
(41)x(31)
For the goodnessoffit test, the sum of the expected frequencies must equal:
n
For a multinomial experiment, what type of test is used to determine whether the sample proportions differ significantly from the hypothesized population proportions?
Goodnessoffit
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